Technical Summary: p-Adic Maass–Shimura Operators on $\mu$-Ordinary Igusa Varieties

1. Problem Statement

The paper addresses the problem of $p$-adically interpolating algebraic Maass–Shimura differential operators on Shimura varieties of Hodge type. While classical Maass–Shimura operators act on nearly holomorphic automorphic forms and are crucial for studying special values of $L$-functions, their $p$-adic interpolation has historically been limited. Previous constructions (e.g., by Eischen–Fintzen–Mantovan–Varma and Eischen–Mantovan) successfully interpolated these operators in specific cases, such as unitary Shimura varieties, but often relied on explicit coordinate descriptions (Serre–Tate coordinates) and were restricted to specific subalgebras of differential operators or specific loci (the ordinary locus).

The authors aim to generalize these constructions to the $\mu$-ordinary locus of general Hodge-type Shimura varieties. Specifically, they seek to:

1. Construct a systematic, coordinate-free framework for these operators on $\mu$-ordinary Igusa varieties.
2. Integrate the action of rank-one differential operators into an action of an explicit formal group.
3. Use $p$-adic Fourier theory to extend this action to an algebra of functions on the Tate module of a dual $p$-divisible group, thereby achieving a full $p$-adic interpolation.
4. Clarify the relationship between these differential operators and $p$-adic Hecke operators.

2. Methodology

The authors employ a synthesis of integral models of Shimura varieties, crystalline Dieudonné theory, and $p$-adic Fourier theory.

2.1. Geometric Setup and Canonical Torsors

• Integral Models: They work with the canonical integral model $S_K$ of a Shimura variety of Hodge type $(G, X)$ over the ring of integers of the reflex field.
• Canonical Torsor ($P_{dR}$): They utilize the canonical $P_{\mu^{-1}}$-torsor $P_{dR}$ over $S_K$, where $P_{\mu^{-1}}$ is the parabolic subgroup associated with the inverse of the Hodge cocharacter $\mu$. The space of algebraic $p$-integral nearly holomorphic automorphic forms is identified with the global sections $\mathcal{O}(P_{dR})$.
• Parallelization: A key technical innovation is the use of the Gauss–Manin connection to canonically parallelize the tangent bundle $T P_{dR}$. This yields an isomorphism $T P_{dR} \cong \mathfrak{g} \otimes \mathcal{O}_{P_{dR}}$, where $\mathfrak{g}$ is the Lie algebra of $G$. This allows the Lie algebra $\mathfrak{u}_\mu$ (the Lie algebra of the unipotent radical of the opposite parabolic) to act on $\mathcal{O}(P_{dR})$ as algebraic Maass–Shimura operators.

2.2. Igusa Varieties and Formal Groups

• $\mu$-Ordinary Locus: The authors focus on the $\mu$-ordinary locus, where the Newton polygon is minimal. They consider the Mantovan Igusa variety $Ig_M$, which is a pro-étale torsor for the group $M_\mu(\mathbb{Z}_p)$ (the centralizer of $\mu$) over the $\mu$-ordinary locus.
• Construction of $H$: By analyzing the automorphism group of the canonical lift of the $\mu$-ordinary $p$-divisible group, they construct an explicit $p$-divisible formal group $H$ over the base. The Lie algebra of $H$ is identified with the $N$-invariants of the unipotent Lie algebra, $\mathfrak{u}_\mu^N$ (where $N = M_\mu \cap U_\mu$).
• Integration: They prove that the action of the Lie algebra $\text{Lie}(H)$ on the space of $p$-adic automorphic forms $\mathcal{O}(Ig_M)$ integrates to an action of the formal group $H$ itself.

2.3. $p$-Adic Fourier Theory

• Dual Action: Using the integral $p$-adic Fourier theory developed in [27], the action of the formal group $H$ is dualized to an action of the algebra of functions on the Tate module of the Serre dual, $\mathcal{O}(T_p H^\vee)$.
• Locally Analytic Functions: By passing to the generic fiber and inverting $p$, they describe the action in terms of an algebra of locally analytic functions. They introduce a period automorphism $\Omega_\zeta$ (dependent on a choice of trivialization of $\mathbb{Z}_p(1)$) to relate the algebraic operators to these analytic functions.

3. Key Contributions and Results

3.1. Main Theorems

• Theorem A (Formal Group Action): There exists an explicit $p$-divisible formal group $H$ with an $M_\mu(\mathbb{Z}_p)$-equivariant Lie algebra isomorphic to $\mathfrak{u}_\mu^N$. This group acts on the Mantovan Igusa variety $Ig_M$, and the differentiation of this action recovers the algebraic Maass–Shimura operators.
• Corollary 1.2.7 (Integral Interpolation): The action of the symmetric algebra $\text{Sym}(\mathfrak{u}_\mu^N)$ extends to a continuous action of the algebra of functions $\mathcal{O}(T_p H^\vee)$ on $\mathcal{O}(Ig_M)$. This provides a $p$-adic interpolation of the differential operators.
• Corollary 1.2.11 (Locally Analytic Interpolation): After inverting $p$, the action extends to an algebra of $\gamma$-locally analytic functions $\mathcal{O}_{\gamma\text{-la}}(u_{\mu,0}^*, C)$. This generalizes previous results to the $\mu$-ordinary case and higher dimensions.
• Theorem B (Near Overconvergence): In the ordinary case ($E = \mathbb{Q}_p$), the action extends to the space of nearly overconvergent automorphic forms, generalizing results by Andreatta–Iovita–Stevens and others.
• Theorem C (Relation to Hecke Operators): The paper establishes a precise compatibility between the differential operator action and the Hecke action. Specifically, the action of the subalgebra of locally constant functions on the Tate module corresponds exactly to the action of the Hecke algebra $C[U_{\mu,0}(\mathbb{Q}_p)/U_{\mu,0}(\mathbb{Z}_p)]$. This unifies the differential and Hecke actions within a single algebraic framework.

3.2. Technical Innovations

• Coordinate-Free Construction: Unlike previous works that relied on explicit Serre–Tate coordinates, this paper uses the parallelization of the tangent bundle of $P_{dR}$ via the Gauss–Manin connection. This simplifies the theory and makes the composition of operators transparent.
• Generalization to $\mu$-Ordinary: The construction works for the $\mu$-ordinary locus, which is strictly larger than the ordinary locus in non-PEL or non-split cases, and handles cases where the unipotent radical is non-abelian.
• Period Automorphism: The introduction of the period automorphism $\Omega_\zeta$ allows for a precise matching of the $p$-adic interpolation with the $E'$-rational Maass–Shimura operators, correcting for the difference between the Hodge filtration and the slope filtration.

4. Significance and Claims

The authors claim that their results:

1. Optimally Extend prior constructions of $p$-adic Maass–Shimura operators to the $\mu$-ordinary setting for general Hodge-type Shimura varieties.
2. Clarify the relationship between differential operators and Hecke operators, showing that they are intertwined via the action of a single algebra of locally analytic functions.
3. Recover known results (such as those for the ordinary case and unitary Shimura varieties) as special cases of their general framework.
4. Provide a Foundation for the construction of $p$-adic $L$-functions. By interpolating the differential operators, the authors suggest a pathway to constructing $p$-adic $L$-functions in settings where the $\mu$-ordinary locus is not the ordinary locus, potentially clarifying choices of test vectors in existing constructions.

The paper explicitly notes that while it does not prove a $\mu$-ordinary generalization of the near overconvergence result (Theorem B) for all cases, it formulates an optimistic conjecture (Conjecture 6.4.2) regarding this extension. The work is presented as a systematic theoretical advancement rather than an immediate computational tool, emphasizing the structural unification of automorphic forms, differential operators, and $p$-adic analysis.